A set of middle school student heights are normally distributed with a mean of $150$ centimeters and a standard deviation of $20$ centimeters. Let $X$ represent the height of a randomly selected student from this set. Find $P(X<115)$. You may round your answer to two decimal places.
Representing probability with area Since we know the distribution of heights is normally distributed, the probability $P(X<115)$ can be found by calculating the shaded area below $X=115$ in the corresponding normal distribution: $90$ $110$ $130$ $150$ $170$ $190$ $210$ $ \mu_X = 150$ $ \sigma_X = 20$ $ P(X<115)$ $115$ Calculating shaded area We can use the "normalcdf" function on most graphing calculators to find the shaded area: $\begin{aligned} &\text{normalcdf:} \\\\ &\text{lower bound: } -9999 \\\\ &\text{upper bound: } 115 \\\\ &\mu=150 \\\\ &\sigma=20 \end{aligned}$ Output: $\approx0.040$ [Why do we use normalcdf instead of normalpdf?] Answer $P(X>175) \approx 0.04$ [What if I don't have a fancy calculator?]